Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{q^2 - 36}{q + 6}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = q$ $ b = \sqrt{36} = 6$ So we can rewrite the expression as: $y = \dfrac{({q} + {6})({q} {-6})} {q + 6} $ We can divide the numerator and denominator by $(q + 6)$ on condition that $q \neq -6$ Therefore $y = q - 6; q \neq -6$